Concentration anisotropy and directionality in the dielectric breakdown problem on a square lattice

Abstract

The authors propose a small-cell position-space renormalisation group approach to study the effects of concentration anisotropy and directionality in the dielectric breakdown problem on a two-dimensional square lattice. They show that, in spite of the anisotropies inherent in the problem, concentration anisotropy is irrelevant (in the renormalisation group sense) both for the undirected and directed cases, just as occurs in normal percolation. For respective bond concentrations x and y along the Cartesian directions, they remark on relevant features of the phase diagram of insulating and conducting phases in the xy plane. For x=y=p, and V_b the breakdown voltage, the phase diagram in the p-V_b plane displays a line of fixed points on the p=1 axis, whose physical meaning is discussed. Directional effects are included via 'directed insulators', an approximation which results in the appearance of a mixed (non-conducting) phase consisting of conductors and 'mismatched' directed insulators.